Consider the following multivariate polynomial system, which defines $x(s)$ and $y(s)$,
$$s - y \, ( s x + x - 1) = 0$$ $$s - x \, ( s y + y - 1) = 0$$
One may describe $x$ and $y$ as functions of $s$ by finding their implicit equations using any technique of polynomial elimination,
$$\Phi_1(x,s) = (s+1) \, x^2 - x - s = 0$$
$$\Phi_2(y,s) = (s+1) \, y^2 - y - s = 0$$
Implicit function theorem implies that there is a pole candidate at $s=-1$.
That is my question: there is a general way to study the poles of an algebraic function defined by a multivariate polynomial system without explicitly obtaining its implicit equation? The whole point here is trying to skip the time-consuming elimination process (to the best of my knowledge only algorithms based on Resultants or Gröbner Basis can be used).